S-duality and a large N phase transition in N=4 SYM on K3 at strong coupling

We study the supersymmetric partition function of {\cal N}= 4 super Yang-Mills with gauge group SU(N) on K3 in the large N, fixed g limit and show that it undergoes a first order phase transition at the S-duality invariant value of the gauge coupling g. Turning on the \theta-angle we find lines of phase transitions on the \tau plane. The resulting phase diagram and the large N free energy are exactly SL(2,Z) invariant. Similar phase transitions take place in systems related to the {\cal N}=4 on K3 by dualities. One of them is the Dabholkar-Harvey heterotic string system. We consider its mixed (a la Ooguri-Strominger-Vafa) partition function allowing contributions from multi-string states. We find that in the large winding charge limit, it undergoes a phase transition with respect to chemical potential for momentum. It is a short-string, long-string transition that we find interesting in connection with black hole entropy counting.